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<h1>v_rotro2qr
</h1>

<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="box"><strong>V_ROTRO2QR converts a 3x3 rotation matrix to a real quaternion</strong></div>

<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="box"><strong>function q=v_rotro2qr(r) </strong></div>

<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="fragment"><pre class="comment">V_ROTRO2QR converts a 3x3 rotation matrix to a real quaternion
 Inputs: 

     R(3,3)   Input rotation matrix

 Outputs: 

     Q(4,1)   normalized real-valued quaternion

 In the quaternion representation of a rotation, and q(1) = cos(t/2) 
 where t is the angle of rotation in the range 0 to 2pi
 and q(2:4)/sin(t/2) is a unit vector lying along the axis of rotation
 a positive rotation about [0 0 1] takes the X axis towards the Y axis.</pre></div>

<!-- crossreference -->
<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
This function calls:
<ul style="list-style-image:url(../matlabicon.gif)">
<li><a href="v_rotqr2ro.html" class="code" title="function r=v_rotqr2ro(q)">v_rotqr2ro</a>	ROTQR2RO converts a real quaternion to a 3x3 rotation matrix</li></ul>
This function is called by:
<ul style="list-style-image:url(../matlabicon.gif)">
<li><a href="v_rectifyhomog.html" class="code" title="function [imr,xa,ya]=v_rectifyhomog(ims,roc,k0,mode)">v_rectifyhomog</a>	V_RECTIFYHOMOG Apply rectifying homographies to an image set</li><li><a href="v_rotpl2ro.html" class="code" title="function r=v_rotpl2ro(u,v,t)">v_rotpl2ro</a>	V_ROTPL2RO find matrix to rotate in the plane containing u and v r=[u,v,t]</li><li><a href="v_rotro2eu.html" class="code" title="function e=v_rotro2eu(m,r)">v_rotro2eu</a>	V_ROTRO2EU converts a 3x3 rotation matrix into the corresponding euler angles</li><li><a href="v_rotro2pl.html" class="code" title="function [u,v,t]=v_rotro2pl(r)">v_rotro2pl</a>	V_ROTRO2PL find the plane and rotation angle of a rotation matrix [u,v,t]=r</li></ul>
<!-- crossreference -->


<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function q=v_rotro2qr(r)</a>
0002 <span class="comment">%V_ROTRO2QR converts a 3x3 rotation matrix to a real quaternion</span>
0003 <span class="comment">% Inputs:</span>
0004 <span class="comment">%</span>
0005 <span class="comment">%     R(3,3)   Input rotation matrix</span>
0006 <span class="comment">%</span>
0007 <span class="comment">% Outputs:</span>
0008 <span class="comment">%</span>
0009 <span class="comment">%     Q(4,1)   normalized real-valued quaternion</span>
0010 <span class="comment">%</span>
0011 <span class="comment">% In the quaternion representation of a rotation, and q(1) = cos(t/2)</span>
0012 <span class="comment">% where t is the angle of rotation in the range 0 to 2pi</span>
0013 <span class="comment">% and q(2:4)/sin(t/2) is a unit vector lying along the axis of rotation</span>
0014 <span class="comment">% a positive rotation about [0 0 1] takes the X axis towards the Y axis.</span>
0015 
0016 <span class="comment">%      Copyright (C) Mike Brookes 2007-2018</span>
0017 <span class="comment">%      Version: $Id: v_rotro2qr.m 10865 2018-09-21 17:22:45Z dmb $</span>
0018 <span class="comment">%</span>
0019 <span class="comment">%   VOICEBOX is a MATLAB toolbox for speech processing.</span>
0020 <span class="comment">%   Home page: http://www.ee.ic.ac.uk/hp/staff/dmb/voicebox/voicebox.html</span>
0021 <span class="comment">%</span>
0022 <span class="comment">%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</span>
0023 <span class="comment">%   This program is free software; you can redistribute it and/or modify</span>
0024 <span class="comment">%   it under the terms of the GNU General Public License as published by</span>
0025 <span class="comment">%   the Free Software Foundation; either version 2 of the License, or</span>
0026 <span class="comment">%   (at your option) any later version.</span>
0027 <span class="comment">%</span>
0028 <span class="comment">%   This program is distributed in the hope that it will be useful,</span>
0029 <span class="comment">%   but WITHOUT ANY WARRANTY; without even the implied warranty of</span>
0030 <span class="comment">%   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the</span>
0031 <span class="comment">%   GNU General Public License for more details.</span>
0032 <span class="comment">%</span>
0033 <span class="comment">%   You can obtain a copy of the GNU General Public License from</span>
0034 <span class="comment">%   http://www.gnu.org/copyleft/gpl.html or by writing to</span>
0035 <span class="comment">%   Free Software Foundation, Inc.,675 Mass Ave, Cambridge, MA 02139, USA.</span>
0036 <span class="comment">%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</span>
0037 
0038 <span class="comment">% in the comments below, t is the rotation angle, a is the rotation axis</span>
0039 q=zeros(4,1);
0040 t = 1 + r(1) + r(5) + r(9);     <span class="comment">% 2(1+cos(t)) = 4(cos(t/2))^2 = 4 q(1)^2</span>
0041 <span class="keyword">if</span> t&gt;1                       <span class="comment">% for rotation angles less than 120 degrees</span>
0042     s = sqrt(t) * 2;            <span class="comment">% 4 cos(t/2) = 2 sin(t)/sin(t/2)</span>
0043     q(2) = ( r(6) - r(8) ) / s;
0044     q(3) = ( r(7) - r(3) ) / s;
0045     q(4) = ( r(2) - r(4) ) / s;
0046     q(1) = 0.25 * s;            <span class="comment">% cos(t/2)</span>
0047 <span class="keyword">elseif</span> ( r(1) &gt; r(5) &amp;&amp; r(1) &gt; r(9) )            <span class="comment">% row 1</span>
0048     s  = sqrt( 1.0 + r(1) - r(5) - r(9) ) * 2;  <span class="comment">% 4 a(1) sin(t/2) = 2 a(2) (1-cos(t))/sin(t/2)</span>
0049     q(2) = 0.25 * s;
0050     q(3) = (r(2) + r(4) ) / s;
0051     q(4) = (r(7) + r(3) ) / s;
0052     q(1) = (r(6) - r(8) ) / s;
0053 <span class="keyword">elseif</span> ( r(5) &gt; r(9) )                           <span class="comment">% row 2</span>
0054     s  = sqrt( 1.0 + r(5) - r(1) - r(9) ) * 2;
0055     q(2) = (r(2) + r(4) ) / s;
0056     q(3) = 0.25 * s;
0057     q(4) = (r(6) + r(8) ) / s;
0058     q(1) = (r(7) - r(3) ) / s;
0059 <span class="keyword">else</span>                                             <span class="comment">% row 3</span>
0060     s  = sqrt( 1.0 + r(9) - r(1) - r(5) ) * 2;
0061     q(2) = (r(7) + r(3) ) / s;
0062     q(3) = (r(6) + r(8) ) / s;
0063     q(4) = 0.25 * s;
0064     q(1) = (r(2) - r(4) ) / s;
0065 <span class="keyword">end</span>
0066 q=q*(2*(q(find(q~=0,1))&gt;0)-1); <span class="comment">% force leading coefficient to be positive</span>
0067 <span class="keyword">if</span> abs(q'*q-1)&gt;1e-8
0068     error(<span class="string">'Input to rotro2qr must be a rotation matrix with det(r)=+1'</span>);
0069 <span class="keyword">end</span>
0070 <span class="keyword">if</span> ~nargout
0071     <a href="v_rotqr2ro.html" class="code" title="function r=v_rotqr2ro(q)">v_rotqr2ro</a>(q); <span class="comment">% plot a cube</span>
0072 <span class="keyword">end</span>
0073</pre></div>
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